Accounting for Correlations When Fitting Extra Cosmological Parameters. (arXiv:1904.10521v1 [astro-ph.CO])
<a href="http://arxiv.org/find/astro-ph/1/au:+Huang_Y/0/1/0/all/0/1">Yajing Huang</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Addison_G/0/1/0/all/0/1">Graeme Addison</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Bennett_C/0/1/0/all/0/1">Charles Bennett</a>
Current cosmological tensions motivate investigating extensions to the
standard $Lambda$CDM model. Additional model parameters are typically varied
one or two at a time, in a series of separate tests. The purpose of this paper
is to highlight that information is lost by not also examining the correlations
between these additional parameters, which arise when their effects on model
predictions are similar, even if the parameters are not varied simultaneously.
We show how these correlations can be quantified with simulations and Markov
Chain Monte Carlo (MCMC) methods. As an example, we assume that $Lambda$CDM is
the true underlying model, and calculate the correlations expected between the
phenomenological lensing amplitude parameter, $A_L$, the running of the
spectral index, $n_{rm run}$, and the primordial helium mass fraction, $Y_P$,
when these parameters are varied one at a time along with the $Lambda$CDM
parameters in fits to the $textit{Planck}$ 2015 temperature power spectrum.
These correlations are not small, ranging from 0.31 ($A_L-n_{rm run}$) to
$-0.93$ ($n_{rm run}-Y_P$). We find that the values of these three parameters
from the $textit{Planck}$ data are consistent with $Lambda$CDM expectations
within $0.9sigma$ when the correlations are accounted for. This does not
explain the 1.8-2.7$sigma$ $textit{Planck}$ preference for $A_L>1$, but
provides an additional $Lambda$CDM consistency test. For example, if $A_L>1$
was a symptom of an underlying systematic error or some real but unknown
physical effect that also produced spurious correlations with $n_{rm run}$ or
$Y_P$ our test might have revealed this. We recommend that future cosmological
analyses examine correlations between additional model parameters in addition
to investigating them separately, one a time.
Current cosmological tensions motivate investigating extensions to the
standard $Lambda$CDM model. Additional model parameters are typically varied
one or two at a time, in a series of separate tests. The purpose of this paper
is to highlight that information is lost by not also examining the correlations
between these additional parameters, which arise when their effects on model
predictions are similar, even if the parameters are not varied simultaneously.
We show how these correlations can be quantified with simulations and Markov
Chain Monte Carlo (MCMC) methods. As an example, we assume that $Lambda$CDM is
the true underlying model, and calculate the correlations expected between the
phenomenological lensing amplitude parameter, $A_L$, the running of the
spectral index, $n_{rm run}$, and the primordial helium mass fraction, $Y_P$,
when these parameters are varied one at a time along with the $Lambda$CDM
parameters in fits to the $textit{Planck}$ 2015 temperature power spectrum.
These correlations are not small, ranging from 0.31 ($A_L-n_{rm run}$) to
$-0.93$ ($n_{rm run}-Y_P$). We find that the values of these three parameters
from the $textit{Planck}$ data are consistent with $Lambda$CDM expectations
within $0.9sigma$ when the correlations are accounted for. This does not
explain the 1.8-2.7$sigma$ $textit{Planck}$ preference for $A_L>1$, but
provides an additional $Lambda$CDM consistency test. For example, if $A_L>1$
was a symptom of an underlying systematic error or some real but unknown
physical effect that also produced spurious correlations with $n_{rm run}$ or
$Y_P$ our test might have revealed this. We recommend that future cosmological
analyses examine correlations between additional model parameters in addition
to investigating them separately, one a time.
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